Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D82 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C210 |
| direct product | C2×D82 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | D42 |
| commutator, Frattini, cosocle | C42 | C25 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
2762 conjugacy classes of subgroups; only the 610 normal subgroups are shown.
This group has 2762 conjugacy classes of subgroups — too many to draw. Only its 610 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 18 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 18 points:
(1,2);(3,4,5,6,7,8,9,10);(4,10)(5,9)(6,8);(11,12,13,14,15,16,17,18);(12,18)(13,17)(14,16)98 conjugacy classes — view the full character table.
| maximal subgroup of | #72214276, #72214525, #72214663, … |
| maximal normal subgroup of | #72214276, #72214525, #72214663, … |
| maximal quotient of | #72214134, C2×#492532, C4×D82, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6265552);